Finite Group Databasebeta

← #29689687

Subgroups — #29689687

Order 15616.

1502 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840309C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#384×D61C211yesmaximal, characteristic, complemented
4#21446778C212yesmaximal, complemented
5#21446623C212yesmaximal, complemented
6#21446135C212yesmaximal, complemented
7#21446133C212yesmaximal, complemented
8#21445820C211yesmaximal, characteristic, complemented
9#21445819C211yesmaximal, characteristic, complemented
10#21445816C211yesmaximal, characteristic, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840309—24no—
14#16840309C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519752C2211yescharacteristic
19C2×#14519750—21no—
20C2×#14519748C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×C22⋊D61—21no—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839814—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519699C2212yes—
30C2×#14519698C2212yes—
31C2×#14519697C2212yes—
32C2×#14519689C2212yes—
33C2×C8×Dic61C2212yes—
34#16839632C2211yescharacteristic
35#16839631—21no—
36#16839631—21no—
37#16839631C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758222—28no—
47C2×#11758222—44no—
48C2×#11758222—44no—
49C22×Dic122—21no—
50C22×D244—21no—
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—22no—
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C22×#92418C2312yes—
61#14519776—22no—
62#14519776C2×C414yescomplemented
63#14519773—22no—
64#14519773C2×C414yescomplemented
65#14519763—28no—
66#14519752—22no—
67#14519752C2×C414yes—
68#14519750—28no—
69#14519748—22no—
70#14519748C2×C414yes—
71C4×D244—22no—
72C4×D244C2×C414yescomplemented
73C4×C22⋊D61—28no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2312yes—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—22no—
80C2×#11758196—22no—
81C2×#11758196—22no—
82C4⋊C4×D61—42no—
83C4⋊C4×D61—42no—
84C4⋊C4×D61—44no—
85C4⋊C4×D61—44no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92C42×D61C2×C414yes—
93C42×D61D414yes—
94C4⋊#92418—22no—
95#14519699—42no—
96#14519698—42no—
97#14519697—42no—
98#14519689—24no—
99C8×Dic61—24no—
100C22×D4×C61C2311yescharacteristic
101C23×C4×C61—21no—
102C22×C8×C61C2312yes—
103C4×D4×C61—22no—
104C4×D4×C61C2×C414yescomplemented
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C22⋊C4×C61—21no—
110C2×C42×C61C2311yescharacteristic
111C4⋊C8×C61—22no—
112#102×C61—42no—
113C24×C61—21no—
114C23×D61—21no—
115C23×D61C2×D411yescharacteristic
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61C22×C412yes—
122C22×C4×C61—21no—
123C22×C4×C61—24no—
124C22×C4×C61—22no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—24no—
131C2×C22⋊D61—24no—
132C2×C22⋊D61—24no—
133C2×C22⋊D61—42no—
134C2×C22⋊D61—42no—
135C2×C22⋊D61—42no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144#11758222—432no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—24no—
150C2×D244—21no—
151C2×D244—42no—
152C2×D244—42no—
153C2×D244C22×C412yes—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C4×D61C22⋊C414yes—
167C2×C4×D61C22⋊C414yes—
168C2×C4×D61D814yes—
169C2×C4×D61SD1614yes—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C2×D412yes—
172C2×Dic122—24no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—24no—
185C42×C61C22×C412yes—
186C42×C61C2×D412yes—
187C22⋊Dic61—24no—
188#11758198—24no—
189#11758198—24no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61C22×C412yes—
196#11758196—24no—
197#11758196—44no—
198#11758196—44no—
199#11758196—44no—
200C4×Dic61C22⋊C414yes—
201C4×Dic61C22⋊C414yes—
202C4×Dic61C22×C412yes—
203C4×Dic61C2×D412yes—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—22no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61C22×D411yescharacteristic
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61C2×C22⋊C412yes—
227C22×D61C2×D812yes—
228C22×D61C2×SD1612yes—
229D4×C61—22no—
230D4×C61—22no—
231D4×C61—41no—
232D4×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—48no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×D812yes—
245C2×C4×C61C2×SD1612yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C22⋊D61—48no—
253C22⋊D61—48no—
254C22⋊D61—48no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×D812yes—
267C2×Dic61C2×SD1612yes—
268C2×Dic61AGL(1,ℤ/8)12yes—
269C2×Dic61#13712yes—
270D244—22no—
271D244—22no—
272D244—48no—
273D244—41no—
274D244—41no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—28no—
280C4×D61—48no—
281C4×D61—88no—
282C4×D61#102116yescomplemented
283Dic122—48no—
284C8×C61—42no—
285#92418—42no—
286#21859—611nomaximal, Sylow
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61C22×C22⋊C411yescharacteristic
295C22×C61#53511yescharacteristic
296C22×C61#53611yescharacteristic
297C22×C61C22×D811yescharacteristic
298C22×C61C22×SD1611yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61C2×#10214yes—
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—82no—
318C4×C61C2×#10214yes—
319C4×C61#52011yescharacteristic
320C4×C61#52011yescharacteristic
321C4×C61#53711yescharacteristic
322C4×C61#53711yescharacteristic
323Dic61—22no—
324Dic61—42no—
325Dic61—82no—
326Dic61C2×#10214yes—
327Dic61#41714yes—
328C2×#532—611no—
329C2×#431—611no—
330C2×#384—611no—
331#1990—612no—
332#1957—612no—
333#1874—612no—
334#1874—612no—
335#1802—611no—
336#1802—611no—
337#1802—611no—
338#1802D6111yescharacteristic, complemented
339C2×C61—82no—
340C2×C61#231111yescharacteristic
341C2×C61C22×#10211yescharacteristic
342C2×C61C2×#41711yescharacteristic
343C2×C61#320511yescharacteristic
344C2×C61#320611yescharacteristic
345C2×C61#322611yescharacteristic
346C2×C61#322711yescharacteristic
347D61—24no—
348D61—82no—
349C22×#41—1221no—
350#532—618no—
351#532—1224no—
352C2×C4⋊Q8—611no—
353C2×#127—611no—
354C2×#124—1221no—
355C2×#121—611no—
356C2×#121—611no—
357C2×#121—1221no—
358C2×C4×D4—611no—
359C2×C4×D4—1221no—
360C2×C4×D4C2×D6111yescharacteristic
361C22×C4⋊C4—612no—
362C22×C4⋊C4—1221no—
363C22×C4⋊C4—1221no—
364C22×C42—611no—
365#431—1224no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8C2×D6111yescharacteristic
368C2×#102—612no—
369C2×#102—612no—
370C2×#102—612no—
371C2×#102C2×D6112yes—
372C2×C22⋊C8—612no—
373C2×C4×C8—612no—
374#384—612no—
375#384—611no—
376#384—1221no—
377#384—1221no—
378#384C2×D6111yescharacteristic
379#340—612no—
380#340—612no—
381#340—612no—
382#340C2×D6112yes—
383C61#2185911yescharacteristic, complemented, Sylow
384C2×#41—1228no—
385C2×#41—2444no—
386C2×#41—2444no—
387C22×Q8—1221no—
388C22×D4—611no—
389C22×D4—1221no—
390C22×D4—1221no—
391C22×D4—1221no—
392C22×D4—1221no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—611no—
398C23×C4—1222no—
399C23×C4—1221no—
400C22×C8—612no—
401C22×C8C22×D6112yes—
402C4⋊Q8—614no—
403C4⋊Q8—1222no—
404#127—614no—
405#127—1222no—
406#124—1228no—
407#121—614no—
408#121—614no—
409#121—1228no—
410#121—1222no—
411#121—1222no—
412C4×D4—22no—
413C4×D4—614no—
414C4×D4—1228no—
415C4×D4—1222no—
416C4×D4C4×D6114yescomplemented
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—612no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1222no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—2442no—
428C2×C4⋊C4—2442no—
429C2×C4⋊C4—2444no—
430C2×C4⋊C4—2444no—
431C2×C4⋊C4C22×D6112yes—
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—614no—
441C2×C42—614no—
442C2×C42C22×D6111yescharacteristic
443C4⋊C8—22no—
444C4⋊C8—1222no—
445#102—42no—
446#102—2442no—
447#102—2442no—
448#102—2442no—
449C22⋊C8—1224no—
450C4×C8—1224no—
451C24—21no—
452C24—611no—
453C24—1221no—
454#41—24432no—
455C2×Q8—1224no—
456C2×Q8—2442no—
457C2×Q8—2442no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—612no—
464C2×D4—1224no—
465C2×D4—1224no—
466C2×D4—1224no—
467C2×D4—1224no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—2442no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4—2442no—
479C2×D4—2442no—
480C2×D4—2442no—
481C2×D4C2×C4×D6112yes—
482C22×C4—21no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—614no—
490C22×C4—614no—
491C22×C4—614no—
492C22×C4—611no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—614no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1224no—
504C22×C4—1222no—
505C22×C4—2444no—
506C22×C4—2444no—
507C22×C4—2444no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—1222no—
517C2×C8—1222no—
518C2×C8—1222no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—612no—
525C4⋊C4—1224no—
526C4⋊C4—1221no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—2442no—
530C4⋊C4—2444no—
531C4⋊C4—2444no—
532C4⋊C4—2444no—
533C4⋊C4—2442no—
534C4⋊C4C2×C4×D6112yes—
535C22⋊C4—24no—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—1224no—
539C42—612no—
540C42—612no—
541C42—614no—
542C42—614no—
543C42C2×C4×D6112yes—
544C42D4×D6112yes—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—612no—
553C23—612no—
554C23—612no—
555C23—1221no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1222no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23C2×D4×D6111yescharacteristic
567Q8—2448no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4—1222no—
573D4—1222no—
574D4—2448no—
575D4—2448no—
576D4—2448no—
577D4—2448no—
578D4—2441no—
579D4—2441no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—6116no—
596C2×C4—612no—
597C2×C4—1228no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—4888no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2448no—
612C2×C4C22⋊C4×D6112yes—
613C2×C4C22⋊C4×D6112yes—
614C2×C4D8×D6112yes—
615C2×C4SD16×D6112yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#1451982211yescharacteristic
619C2×C4#1451982311yescharacteristic
620C2×C4#1451982411yescharacteristic
621C2×C4#1451982511yescharacteristic
622C8—42no—
623C8—2442no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—614no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—1222no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22C2×C22⋊C4×D6111yescharacteristic
646C22#1684032811yescharacteristic
647C22#1684033511yescharacteristic
648C22C2×D8×D6111yescharacteristic
649C22C2×SD16×D6111yescharacteristic
650C22AGL(1,ℤ/8)×D6111yescharacteristic
651C22#137×D6111yescharacteristic
652C4—22no—
653C4—42no—
654C4—82no—
655C4—614no—
656C4—614no—
657C4—1222no—
658C4—4882no—
659C4—2442no—
660C4#102×D6114yes—
661C4#1684024211yescharacteristic
662C4#1684024311yescharacteristic
663C4#1684034111yescharacteristic
664C4#1684034211yescharacteristic
665C2—82no—
666C2—1224no—
667C2—4882no—
668C2#2144840811yescharacteristic
669C2C2×#102×D6111yescharacteristic
670C2#417×D6111yescharacteristic
671C2#2145673611yescharacteristic
672C2#2145674311yescharacteristic
673C2#2145693911yescharacteristic
674C2#2145694611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#21859
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839814C2semidirect
3. #384×D61C2semidirect
4. #21446778C2semidirect
5. #21446623C2semidirect
6. #21446135C2semidirect
7. #21446133C2semidirect
8. #21445820C2semidirect
9. #21445819C2semidirect
10. #21445816C2semidirect
11. C61×#1802C2semidirect
14. #16840309C4semidirect
62. #14519776C2×C4semidirect
64. #14519773C2×C4semidirect
72. C4×D244C2×C4semidirect
104. C4×D4×C61C2×C4semidirect
282. C4×D61#102semidirect
338. #1802D61semidirect
383. C61#21859semidirect
416. C4×D4C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168403092
C2×#168398142
#384×D612
#214467782
#214466232
#214461352
#214461332
#214458202
#214458192
#214458162
C61×#18022
#2185961